How can Picard proved that his method was right?

Question

In order to solve initial value problems .. We know that Picard's method is right , but i need to know how can Picard proved this ?

Answer 1

Since $y^\prime(t)=f(t,y(t))$, we have that $\int^x_{x_0}y^\prime(t)dt=\int^x_{x_0}f(t,y(t))dt$. This means that $y(x)=y(x_0)+\int^x_{x_0}f(t,y(t))dt$. We have: $$y_{n+1}(x)=y(x_0)+\int^x_{x_0}f(t,y_n(t))dt\\ \implies y(x)=y(x_0)+\lim_{n\to\infty}\int^x_{x_0}f(t,y_n(t))dt\\ \implies y(x)=y(x_0)+\int^x_{x_0}f(t,y(t))dt$$ This completes the proof.